Torsion modules and differential operators in infinitely many variables
Abstract
This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with respect to the diagonal ideal in the tensor square of the ring. Various notions of torsion modules for an infinitely generated ideal in a commutative ring lead to various notions of differential operators. We discuss differential operators of transfinite orders and differential operators having no global order at all, but only local orders with respect to specific elements of the ring. Many examples are presented. In particular, we prove that every ordinal can be realized as the order of a differential operator acting on the algebra of polynomials in infinitely many variables over a field. We also discuss extension of differential operators to localizations of rings and modules, and to colocalizations of modules.
Cite
@article{arxiv.2505.07739,
title = {Torsion modules and differential operators in infinitely many variables},
author = {Leonid Positselski},
journal= {arXiv preprint arXiv:2505.07739},
year = {2026}
}
Comments
LaTeX 2e with xy-pic; 52 pages, 10 commutative diagrams; v.3: the proof of what is now Lemma 5.1 moved from the deleted Lemma A.1 in arXiv:2504.18460 to this paper; v.5: two paragraphs added at the end of Section 0.6; new Section 0.8, with references, added; other explanations added here and there; new Remark 7.4 inserted; v.6: several misprints corrected; v.7: several misprints corrected